an:04110974
Zbl 0678.22012
Kottwitz, Robert E.
Tamagawa numbers
EN
Ann. Math. (2) 127, No. 3, 629-646 (1988).
00283953
1988
j
22E55 11F70 12G05
Tamagawa number; connected reductive algebraic group; inner twist; Weil conjecture; p-adic Euler-Poincar?? function; orbital integrals; simple trace formula; Hasse principle; Shalika germ; p-adic group; Steinberg representation
The main theorem is that the Tamagawa number \(\tau(G)\) of a connected reductive algebraic group \(G\) defined over a number field \(F\) is invariant under passage from \(G\) to an inner twist. Since it is known that \(\tau(G)=1\) if \(G\) is simply-connected, semisimple and quasisplit over \(F\) this theorem completes the proof of the Weil conjecture: \(\tau (G)=1\) for \(G\) simply-connected, semisimple. A key ingredient in the proof is the \(p\)-adic Euler-Poincar?? function introduced here. The orbital integrals of this function are shown to have remarkable properties. In particular, the function may be used in Arthur's simple trace formula and the formula then reduces to its stable part (this requires the Hasse principle). The stable parts for \(G\) and an inner twist \(G'\) are compared and the formula \(\tau(G)=\tau(G')\) is squeezed out in familiar fashion. The Euler- Poincar?? function is also shown to give a quick proof of Rogawski's theorem on the Shalika germ for the identity element of a \(p\)-adic group. Finally, a theorem attributed to Casselman says that the Euler-Poincar?? function is, up to a given constant, a pseudo-coefficient of the Steinberg representation.
D. Shelstad